Robotics Control System

For LQR, the cost function is $J = \int (x^TQx + u^TRu)dt$, minimize for $Q = I, R = 1$. What is the control input u?
(i) $-Kx$
(ii) $-K$
(iii) $Qx$
(iv) $Ru$

What is the primary effect of a derivative controller?
(i) Increase steady-state error
(ii) Increase rise time
(iii) Reduce overshoot
(iv) Increase error

Explore the impact of proportional (P), Integral (I) and Derivate (D) Control modes, individually and in combination (PI, PD, PID) on system performance.